If \(\epsilon \)  is unknown, [1]} showed that PE achieves a regret upper bounded by \(O(d\sqrt{T\log (T)}+\epsilon \sqrt{d}T\log T\)  with high probability. We notice that the dependency on \(\epsilon \)  has an extra \(\log T\)  factor as compared to the known misspecification result. To avoid adding the \(\log T\)  factor in the term containing \(\epsilon _m\) , we exploit the knowledge of \(\epsilon _m\)  to modify the confidence intervals in PE. We note that following the same analysis in [1]} by replacing \(\epsilon \)  with \(\epsilon +\epsilon _m\)  results in the same concentration of estimated means except for constant factors, which implies the following result.
